{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Continuous Signals\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Standard Signals\n",
    "\n",
    "Certain [signals](https://en.wikipedia.org/wiki/Signal_%28electrical_engineering%29) play an important role in the theory and practical application of [signal processing](https://en.wikipedia.org/wiki/Signal_processing). They emerge from the theory of signals and systems, are used to characterize the properties of linear time-invariant (LTI) systems or frequently occur in practical applications. These standard signals are introduced and illustrated in the following. The treatise is limited to one-dimensional deterministic time- and amplitude-continuous signals."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Complex Exponential Signal\n",
    "\n",
    "The complex exponential signal over time $t$ is defined by the [complex exponential function](https://en.wikipedia.org/wiki/Exponential_function#Complex_plane)\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = e^{s t} \n",
    "\\end{equation}\n",
    "\n",
    "where $s = \\sigma + j \\omega$ denotes the complex frequency with $\\sigma, \\omega \\in \\mathbb{R}$ and $j$ the imaginary unit $(j^2=-1)$. The signal is often used as a generalized representation of harmonic signals. Using [Euler's formula](https://en.wikipedia.org/wiki/Euler's_formula) above definition can be reformulated as\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = e^{(\\sigma + j \\omega) t} = e^{\\sigma t} \\cos(\\omega t) + j e^{\\sigma t} \\sin(\\omega t)\n",
    "\\end{equation}\n",
    "\n",
    "The real/imaginary part of the exponential signal is given by a weighted cosine/sine with angular frequency $\\omega = 2 \\pi f$. For $t>0$, the time-dependent weight $e^{\\sigma t}$ is\n",
    "\n",
    "* exponentially decaying over time for $\\sigma < 0$,\n",
    "* constantly one for $\\sigma = 0$,\n",
    "* exponentially growing over time for $\\sigma > 0$,\n",
    "\n",
    "and vice-versa for $t<0$. The complex exponential signal is used to model harmonic signals with constant or exponentially decreasing/increasing amplitude."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following example illustrates the complex exponential signal and its parameters. The Python module [SymPy](http://docs.sympy.org/latest/index.html) is used for this purpose. It provides functionality for symbolic variables and functions, as well as their calculus. The required symbolic variables need to be defined explicitly before usage. In the example $t$, $\\omega$ and $\\sigma$ are defined as real-valued symbolic variables, followed by the definition of the exponential signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle e^{t \\left(1.0 i \\omega + \\sigma\\right)}$"
      ],
      "text/plain": [
       " t⋅(1.0⋅ⅈ⋅ω + σ)\n",
       "ℯ               "
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "%matplotlib inline\n",
    "sym.init_printing()\n",
    "\n",
    "t, sigma, omega = sym.symbols('t sigma omega', real=True)\n",
    "s = sigma + 1j*omega\n",
    "x = sym.exp(s*t)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now specific values for the complex frequency $s = \\sigma + j \\omega$ are considered for illustration. For this purpose a new signal is defined by substituting both $\\sigma$ and $\\omega$ with specific values. The real and imaginary part of the signal is plotted for illustration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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oHqQUDEVZyO40n/PcmtyFlFJomtKQD3BG+l5E9yGl1F5oW1OOvsFf3RQLKaX12bnJzBVL+J+zHIqFlILKYRmJvBUMn+v1LbQo0Cw05iHvmDzD9WAWUspA95lLPtqzAOcii0AVPEQ8oA6exQ0fUsqEdchNI8i50XD6lwctyogtdJVDO83r5S1ECozcAoXBj8JVXngo+1U8SCmRVao4l3KGo4MsQp1trB+qUCc3w0BKgVrKYf1UpbtxKBZalCZhGyKGmp0X4xNRQkZ9sujDnTAYdeUwLLQocPHbesF4nB0PUUKRYD39hBgW6RAMsggNg7RpO8FEKNdrWEgpDBnVHofxGW6GQRYBFnVdPY4L5Fd7WEgp0Giim9mt8abl+lYWUkoPY04doa3vAB4XWYQmG16yPYGxc72HQZQwEsPSKMZgbnkcgkEWQfaXRM0wfuXq6wZRwhxjqbLIRsqHYBASYPmGOKfqS/S5PHeDe5BSoIrpdlCOPjF7PhQLKQWDS7afuSOKyex8Kg9SSgtjUC91LqfCvrgoFlLK4HTXZRZDi+VaD8VCpHDREhM5a6+06ePR8R6kFK6qV11KSe3ujB6klAlLjQOhcYe39dkOxUKkwN7MjAaV+EnYVmkPXAcBodC4SyNBzL7RGNO2GC5ECq0DmC3jFUd5c7TttQcXAqXC5KHpNl8NFg9coPa8iIeQAHuN29iRkdnwq8YTwOAiShjwlDDWCvtfG9vp8hAS8hu6FQpMlnkCvs8zbD1ECQ2fHD8TuR0an80uD9DiMNk4xxVYB4ze3KU/5CyMj0VdOmUJiZbeU9oCWpxRb1GkITAE7pQ3iBJKpPf/QsehAbJNZA9RApeQGBU15JDOVggeogSGqHLFhvsktY52CAZRQoejg2mQ45K7O+UQDKKEERNMRgnUwhdvF8Egi9ALzyFkOqGF22eH8IkoYWLmwXildmmMpjkEg5BQ34GeJKWpjbGnewdYxWfjIi/F3I+sV/lPRAlw7CbUFJVqjeH0CwdRQqoMPRdxhLIbh2AQJeScud4qarvlmg/BIErgHDi14ngts3rIIjSGMog4zdPMFtDiUOoM0KSU4+oqbxAlwNrLQ8QJamf7+R6yCHBrhzbOyMc89xAldMwYQ14Mk84OfXSRReidkXDsXpgMr1f4BLQ45pUhLVPgRuyZ1AFW8Zm4j8XePlO53vcT0OKz5tqkHXuA+3GVNwgJXA4eaBoZr/jO+20toMWh9usUBcLQ2r1s4iFKwDejZ815L5Y84iEYRAmZs7X8DsZtuh7oE9DiXEpB78tcZGthr1B4iBLgJPGQAcSw6uv9QAZRwiiBwcyFzlK6K/gEWJxbRtBjkRPESJk9cpV3ECUUTOiQFglR3uaIA2hxGD/oH3JEaTA28pQ3CAn4ullGKiajBJNuP78FWHzCPImMGWyYLAo0+W5hBwGhVS6wYS5/xR64gfSE8roICfDS10Jw5vm2FONThQuB0nkYJIhlhgm+cB56dKMLkcKwK9pjEvUyRhhhUzyIFO6gwPvGTzV0zjrSNnVcSCkdXZFuAi0zHcCbYiFSMmNAMFnLUk2cMpUtigcpJaF6WXlhZGaqe4nFhZTCJuL6f2eMd7vq+AS0eGUIUlajtJa+/RoXUgpmeuplynkgZfcUF1IKLKzChS3IR5659EOx0KJUnv/SPcmayt6lcSGlwCukaZP4QUub4Xp5Cy1Kg24pIoehDbV9UQxESpFjNLWJHB11x+z70KL0wMAtkY8Qtk/kQkqJ3EwTMT5JeOJlXGQROqdZEaNv77V5D1FCihK2SXlJKV7PZJBF4BkDfTt0pXq/hYUWZYwqS3SQ95C2ge9CSsm51/V+BfPbnghcaFH64LxGOSMX5/XuFlIKDEPadyKPqeSrES20KLDdRlrycqwuF1IKbDYJYxR5KNtGcKFF4aGqvN4RttBVi4WUAqetPh8fE/y4+oqFFqWO+rRwjfdzGUQJjE6WCIFMm2acoeVBi9JkLpJ+1+rZEnAhpYwInaE/JcdCr1ostCgVU80aci2fDSAXUorsy+unj4PHWg7FQosCk27qpw+Dy5UXxUCkMNqhB+4mUUnRg93f2IOUAo+rTF0gaLAs8mEYRAlpNp6RoBhmebuqMIgSSoh9qrKNkZGqh2EhpXAMJJ028NgjzkOxkFJgjnBttkskebxe4xPQ4lDmEo6LyS/DutiWmQuRAhu1QvcVmWJpzGxjyIWUwuBWfpHWJdLqTNgeRAqdtMJzSPCvYS5w1/GheNAPkmqhoafx/EqhC8nI4YfiQaAMKA+48CW+uEoScp573daFSIHhD0cNc+vk+nwK21XxEBImOvNgFDWnZYy856EcAMUnbRAM/fbiMbQx21bBHkICz4GMUmhQohfFPTocQIvzbDgcnRreDIzbq68eooSRC3dai8RwnogQDyEhvaFc2XEK91xn2aaWhygBz8lI4ML4hhj2FOIhSoCah0EBKbd0zytYQIvD9CxQjQUNg3fcUTMesgjcmysUw81re8vWQ5QAQ4rnCiHOsR116yFKwM/Q3CpUSBjI1ysbZBHQzzE44JigF9R61WAQEjKP2IQsYubIKPsdHGQRYBRwJ7eIQtnraR6ihMjQ2Ewxz1eez+ogizAnD6tSLJPCRfhElJDgZkCJQFw7Rsg4BIMoIccU2Dyc42dMF8EgiwB3L0jFpc95+qqDKAFKKoNNMcyNUA7BIIvAfUsVX+eVXEQJsC4wR4s4zh1T6SKLwNi4ouJxdL6HKKFlcVkL4wnu4h/yVVjO38hDtrmXThxAizMgsST9CrAurk5hkEXgLrp+Z/7gVYFBlAA/vuvvoLsfo8NDFqEXnqSmuM6QrxoMooSZGf0ovRHz8t3GBlmEQVtPxL22b4RPhAR4HjyfJANqnJH5KdaiEeZAkF4yEwZuO6UNogQ4ZVEVCA8vbyPWQ5TAM+RlioqCkX9awEGUUDrmt0ExN+zaVYNBlMBw7CZ6mZu52xPxECUM/Ar3BxiCN1O4CAYhgau7MH47Z5bJBYn9TR1ECZnhZ6+CWZhRaP2U/wS0eOepNdkSgckYzys7CAmYgAbXdF9cnI/1tJoFWLwzvKFFbtHAoo17t8EBWJw7YDRJ6cnNcTaUHOCHL3xfrtFUHgSOhWF08zlL4EOkDC4YJ+575i6h2c8M5UNKmT13Rh8UJvdKoV4UC5EyYb8lRt4mrs9jLsv7wTwIlMjjaV1Wq5i+hD1gGVQ+pJTBl6xiMJZno92RszBPVA56hEyjMpkHIz3lPUgp3BDhchBzV8Em2C/uQkqpEifKVT26cFcdn4AWh9JpjNCBHHpqhn4IFlIKtCejOVKDQ4JumeehWEgpM2h8LORccxoXxUKkwKKDYVlUjh/dm/k+tCgN9ncX+Th93UWUgGlVVw4ZepL2hogPKYVZD7im13h4be6AVR9aFNQoK4eN0ezbZ/QhpWTOl2mtT57IYR9aFIyEovIcZji90YOUUoosqYsc9m2+3sVCSqlBzDSRj1B7OhQLLQozDzWRl4DZ+aYYSCkNFnnRn2K+nTIPxUKLQvu5L/lMd1NaSCkdXmKIS957vGqx0KIwM0FROdTtMwX6kFJkmp7rHXN89nx8aFE6zCutHWZHLde7WEgpsyTOQCJPc6/k+BApsjrNrGnSkZjqZb+LBy1KK7PqA8tq4rwoBlJK1NBC6eE1pVEOxUKL0hlBL8OoH7fQRZSAZ+xL68ByKvF6LAspBUYHpwBqEEwG9WIYRAn0R7kI3Bg72Mr9tSyklKoHJUUZ1n3c1EWUwI7aVEfPOXK9GBZSCg1TZgNotExbzv1QLKSUyQiSKeGoM4+W8qFYiBQGJgSeIeD0xIMLcWtWD1IKw47nmgFz3xGePqQUvFbnknadspnb6qFYiJTKZenJccotb3z407s8iBSxknjIOnFtAVZe2g3pQaQwd0Wg5cYinJ3bbhcPAiXxIPScTOTJIq0+8UIuQkKHscdIZK62Zm50PuUtwOLcguKmAr4gvIwdTecBWrwXblBWhjTkM2E7AIrz3E7NjAJjqELK9Vn5cREl4NF6F9MTdkLeOs5DlIDppUAR18w8Im0/kYcoodORL68KfZb73nJwESUMamFK6yjtfoVPQIvPQjLFWTO87fIGIUFCLlEXpAHd/jkc7gFaHBMCuZUGT9wZN1xECQmfAbqaoVI8YJMPwSBKwJcOTdyFxnSY8xAMsgiwCmBx1Khbx+UifCJKKEliyqos6O04URdZBOYNk98pOY16vbRBlFDZx+VbZEZ2nvKfwCpO01G+RC5h79a6iBIaLKYpz8lVm3F9I4MoAXMAPS8mUp6xt6sVDLIIXECV3+EKY7vKfwBafJRUYO2LtO+9cxdRwsTXpZOH5+RRzasCgyxCLbL2yqMLIZXrGxmEBCYhkVBZ+XRQIXskOMgi1Fy1LXOfV+lvYi0amZ9MfgE9pp0u5yCL0Ng9RNypDy7CJ6IEeF9N3ol69rytBVbxlhgNQzHXOK8HMogScioMz+FYQo8/49hBFmEM5l/haIU/fHqQgyihwD/r0o5joH2uVzCIEmqNDIGvkpN77gnNQ5QA951L2NRolXH9h2AQJeADZ+gZiDlL9KsVDKKEAY1ZkupkeOVXDQZRAubETp3PdchwvcGHnIVpOHJRV6aIfg5uuogSqM2YvqW824Sb0g/BIEqovXDrl1FfbewcQy5CggTMQskwPDfPevSKBbQ4rCtMKTD/Ao9/5lP8E2Bx+KwYSLB2GSQBc3ZbBx5CQueWXMTUw1UedLG5G8xBNAd4YRq3+ZKtT0zwe/5zIVI6Iy4avfDMlERzZ7bzIVIGM0/Sti4MyYrtCU9zESVgrpb94cr3K+VmWIiUiV+SeAvaffAodwCTDymF6YoZ9M5MwSnudA8+pBRuN3HVonZZ4H322n1IKX30wJWqysQcZeeW8iGlzNojczcxugsafU8PLgRKDRLXyy1M9rgG8+ahuJBS4ghyVqUyuVjIz2kkH1JKok7sIofFUrYidyGl8CglF1Qgxx/nXvZzIaXAhOc2FeWwy8sTtu1DSpEoQa29wrreiw0utChzVukVbLocxvVgFlIKmrjmutyRfWLWRRZhcOVBxJ3Hey7CJ6KEnjQohmJM5+1iWGhReuZIE3kN+WoSgyhhwB2RfJVTQjfK1e4WWhRMEF39MGYPGFclFlIK1A13+FVeS75qsdCi9CInraV2qOWrFguRAqOb2zjrp+rZ1PehRWmdk67IS9j5J31IKTGWMtdPxXa0pAstCk/iri4EC3APYRdSCvQ6bAmVp75nWw9RQk5yrEtiOkOsVxUGUQKc1sKTYxw9zCp6VWGhRWEQtsq53TyvV7eQUuQYlL4fzZC9ZOJCSmnQ/l212mRWm34oFlJKbwyn16V52PzhankLKYU78lnlVe++2BQLKWXK9R4ix/QR+vUuFiJFQhJq0KmDa6m7EgdRAvSArHF0uoSx50MwiBKYZYoLzoy+xvS/LT8XUkqHfmJ8Hh14noa/KBYihWGSQ3qpbO/vOB4XUQLd2KQhMY3BqfMwLEQKc9Rxn/olBiK66VF3HkSKpG5i3K5uqJ80gz4ESpOAT555YZ4VTv8Pw0NIqBIyAFdtMDojj+e4tYuQ0N6Doye9uvg8dVsFHkJCp7M50WMYowRv7NlddBElME1V5Bl3eOZpj0IHYHHaVWjVyl0+6Kczaj1ECVCpDA9mLDXsq225e4gSuB0LZd8k/U8tF8EgShgpMJqBOqy1Ywd4iBImjKPZKU683GQegkFIkP1V9Bgxc3iQ6ilvgVV8DG7WVWa7OoeZXEQJCfMbWrAyWVc51r6HKAHqkRkPn645DsEgSiiNWy6Vhy/iSNcTfQJanKH8WX4lwCreU6yHKKHRMOtc95vwxrZ/4CGLgBmNzi0TS6dQr1c2iBKgcJkEiOLYx00wiBJ49BsTT2VygbFDI1xkEaCtuO7XJRNIuWowiBJmxJBVMTp8u76SQRahiYqiGJPUNvM8ZBHgxBT9HWitfL2DQUDgOYosboiIU9kTkYcswpA4yFXx7noeooSYuEIob7ZTEzjyVbjVqU3D7a94/bpBlJBgExf9mT7D/fwGWYTRGK7Nxk9tpwdyESXklhjgLN1rtHS9gUGUUJiNRD7cZFzMVYNBlFCZ0lVGSN85nRy5FmYYjQjjTOn67Q/5KswLh3Rwx7r9CgfQ4r32puoJqmOnq3YRJaBHNa2VOa/3soCHKGHq7SCyGpHD9qY9hASm9m5MqU4Fi1bc72sBLZ64fx/lipF0N5aDKAEqj4tGvG4CbbOtdg9RQi+FWzSyv1H7du88hATJT4+qOS8OBifvGhxECfRD0HG5KZmZg+0QDEJC5qYqQ71QIKQct0vkISQw1JEZhIccgat7Yd8BWLxJQn7eHxaapJPeY8xBSGC2KZ7aiHKCKp4lYg8BYRRZTeRqDRN1lPZE4rkICVUWy3kAgacDYc7sCceFSOErNclFWmgE5n0c1oeU0muUsP3CpHU7A7KLkEDjuEry8sI8nXE+Z3R9SCmVcWh023hDSjo90IWU0vFHWa1g0vNUnhheH1LK5JJWFnmd40QKuRApUDqwSLmvSgcjx2NEu5BSIpwvBu6jeeMIOxGEDymFZpOKGT4drkoMooScUmLWfy74YpxtFeJCSoFzWHlgg6f5YZnG67EspJTK4JumngXzg/ZDsZBSGhRqUWcEJuHO5OFDSun48Mt/odW2Q0ZcSCkjQHfpomMeTG9yKBZaFF65FUXOwwWnG3uQUngFlay+MJK37XN1PrQosyatHL5xizfjEyFB8u9WRkRRzsMqm+FBSoH1LPv2lMMXDRfFQovSJOGciBkRfzE+ESWkGEtqq+55omVcaFE6owvXG6Yd5O9DSsmY8npbX/Gaal1oUYYmWaIcdvG4KQZSCtpU7lSRcxKwgdKhWGhRMPsWrT1zDN0UAykFP8T+KfI4x1EUHrQoQ8JBKIeplcNVi4WUAvPjaS96/YfwCWhxnhuIqgtmzTsu14eUwkNaQ3UBTzbuFRYXAmXyFHKeoSwlxd3MRXEhpfD+0qFybvM+Ids+pBS44QwrlQUlPMe4arGQUuTWGpVz+XAbES6klDZS1Mmm8GqrehgGUQJPDXK9lTsOKZxoUhcihcezOgM/OQnyjP6Tl9uHlFI7Nw4h511L7ayZuBApSe4zYN/Ok4cz2w4ScyFSeKy206eQDcWU9tEkHyJFLRN+lsitvnR9MA8ihX7R4BHQKAf0mHfooXgQb6nlrJSpQJgPnj7V0kYuQoIczWSiZV6kwY22/hAchITK5KmYahhXzVDkR3m5iBKYnAAdjRbvHDte10VIaLKzjoq5GMBYnKe8BbQ4c2FUWXcazESRTnmDKIEZwjGKYRbPHreP6SJKGIXqkoGCjdfqnfKfgBafUzLKMCCv5sfg8AAW77xrildESARyYLKhVd5BlADPhyvejKxr+byvBbQ48xyhS0HaYSOFesobRAnwSrjPzTC8lMJ5AQdRQmXaQEpzm72UU/4T0OLM75LlVziRjuv3DbIIvEVDnjP2vI0+F1ECPi+3wRhpDmPiJhhECSMGZqCgGGMjXI9kkEVokqGDcYfoKY+j6SJKwChF9xVxyrVerWCQReACdRNvj2tI11c1CAm8MbhynqKYOVT3OzjIIrSSlz/JIxL9InwiSog88dzVAZ37uJSLLAJGONwx5hGdZbumLqIEfAAu1Eri0d57OQSDLMKU62jld8KOj3IRJTAQaAwVQ+lc72AQJRQGEyT9FmEnzHGRReD5Wm0e/LG0i/CJKKGGwbQOFHc5tbUJBlkE+Npdu1jci3YeoMVhQ/Mahya5JLYt5CKL0AvXRdnnI085XoRPRAk9i7tJMd6sXR/VIIsw5VMXZqrcsd8eoMV5/FoVG0Oo7t83CAmSpW7KoEVf3LcSeIAWj0wGJ3owhjMOjHwVnq10kU4e38lX8U9ECZjiuZsNNR65TdAOwSBKqEzcrfMEd4NO+U9AizNiBRYHg/KznDnb5Q2ihJlkF75xxhh9XJ/HICBELqbmiJm9DblvYfc4D1ECk8rDEmAW2TmftXQPYHFuX+M7DAYF8V7VPeN4CAmJ6Urw3QbvG53ziev0ABaXOxkijEUeWmYE6vNFPYQE5qGstH2irPCUvcrqQ6Akhl50rhrzXNuA6/Ks9PkQKUzlN3jmWEJk6bs8z+VCpBQuv4q7I2lI447X9CFSeOqPEcoweyNTGrdnxciHlCLpPasscMHN39/XQ5Qg14GmUvS+qHbKfwJanCnhohjiIeWxJ0MPIaHhQTn1UsyFjuf0s4soIYfA8EFZcONi+8WwkFK4gMuFmsJ8rbz65lAspJRaJE0w5aPubTIXUQLXwSRgX24XmKf5PEgpvQa5JaBItoe9welDShli+GqKh5xbvGqxkFLgHMj1DoXXPzPB1KFYiJTOu//khsHC27n6jsbxoUUZkr9Y5K22rUhcSCkR5lTRd2QS+meN0IeUkggMledccjwUCy1KZSupHB8otYtiIKXg06S6AgP6PvvhIkqg+qpxyVtt18tbaFFKl+snpe65b/vyoUWZNA/XG8roPhQDKYVBt2HFOFBvzEOxkFKgkurTWoy4u9reQosyU2dyIsp5P8bVKhZSCiw9bhmvnrfj/HxIKcyCkld2kxT2JSw+pBTudAb9qcivf31kC5Ei+e3yGniwg2LdCtKDlBJHk1BOKoTU5ml9D1IKXEYd3lAnrfU9jbqQUgocxZQf3RaPfvEgpVS5WkM0KPOQnNf3IKUwEb+kkKmyXBMuioWUMkrtXBHjCX3mwrq+mIVIkfjNJj288ABJ3XaTCyml8J6W+dIgYJ7HPhQLKWUw9VGUFZ7ZYjwdxoNA4VXcTWLZGf3cSgz7I7sQKQwfw7fkbdVQ2hN942lKFyKFORH75Ip+kAzKe/ffh0jh0UeuS2Me55LYTiLnIiAUHpgWV2rKXQHbTHYAFk/vCo0Jv2jIxS87eYyLkMDV5cjY8yFrlXHPEB5CQmGqe8a0M38Sr2/cT+QgSuB6f260D+Ftl/NIDqKETotSxPFanHEALc4kp/CoOjNBy51Pu7xBSGA6CWaj4ppW457Ko+U8RAm8Qwf9RJJRYUy3QzCIEuCScOuCm+Y97FSULqIEBmOll+z5j5auCj4BLY4OyaV+xh9ALT1RfS6iBF6PRhtdrpPJ43oggyihN+a3ohgzSj2N4CBKGExxM7nwN6ExwlWDQZSAyY0n4CUW57L+PIQEZpvt3M1jtE+YOyLVRZQQE8PWXhKA1Gsfh2CQRRiJ0UlcvOTKdboIn4gSoER4D5DGRLVnr8BFlCAbjfo7MHHzRTDIIvTIi3ObpPtu9yMZRAnMx0j3fNIP2GHkLqIEWAK8NVvENV1f9RNYxXndb3uqbddHNYgSeDnNWhA+Y/NTrEV7kDARycfWjnXvIYvAm8Kl4THFbnPFAbQ4j3ckkXKXs11PYxAl4IMlrpmxY5183i6yCJNxjyKG6xyu5jIICV1skSi/M9ETt9fkIUrgIurQ4VdLOcrLQZSQNMsY461G3QezXEQJ9CVUnMsopxUcRAmFuUyzrBjQW8uHYBAl4DFTby8JEcL8mg7BIEroJWXdhcjMl30RDKKECaOoaZTMyP0oYAchAQYeQw4HxXAlTnkLaHEoV+7/8YhUg3Yrp7xBlNA0azETNPayrTEHYHGuanEr6NVp3ssFvau8gyiBiVH5HaYEAMR5CAYBQe4cTzxSPCq/SNpt4CEkYIJrvIrixUxOeRxDwUNI4GJ84tmTmLjQG44z5UKk0JxD1XB/uPFR595D9yFQGitHR6Rthj5Gjfh8KhcihUceg1zvwwikuYecA2jxOZl4VwxGmIF5L025ECkM0ipsS67zZB5i3nV4kFLoIdKHlWTq4yxBupBSeK5v6M5nKCezlQ+Rwny9Q5Ih8bKJgInv0QYupJSMNhV3gdGReedI8yGl8E5PphGStOewOtuhWEgp8NplnYDpK0fe650eogQGqgRdF8NMmJ8dYR9SClRW7CpnYEy7Xt5CSqHqZRgG7e28j6S5yCLwhu8kYhjZx5RzIVJ4KlMuckyS9P3swHiIEjBsJJNK4fnsPrd2cKFF4W6viuH+7PVqD1ECkwiJ95p4evBYjC6kFAxsrrqKXN3lTbGQUphtR15QMsmetTYXWpQpefFFzov2rs9lIaXUHtbzQjnVeb2KQZRAvSGhgJTnsxzrQkrpsbagvitDVu5KLLQovCxlPS4s2NPnPUgpsIa5JSffEeNiXi9vIaXAf2hN7xYMvITkehcLkdL4l6z9lPt+T4iRiyxCiyHp6BljniZxECXEphlOOBTaHOdreZBS4HzLajvHWx/7NI8PLQpTuWvfpj7bzrYLKYVXViZ9Q/jY/QwtD1JKbZUXGIuCKnOHV/mQUjpD25IqwZB2TmIfUgqdPxWn0faBNBdRwlz3mfGqCh5zv17eQqQwXUviMY0kt0lhUt4v70FKqaGGqcn1GM+wt0dciJTBhA/cA+BBc/oLu+kdRAkM6JJ9FiYdizsXgQ+RwvN/jRldE1fuGNK/P5cHgUJ3mbend0m6A7M47nHiQqRIeg+5Dzkwy3rbaah8iBQukRReVC7XRcQx90ayC5HCm0ni6AzIZmbB1Lf54UKgyG2KmQv6M9DXSNst9RASOAboVdHkK+MyZz2EhCQ3lsOcH0xZInedL4KDKKF2uXiVVmg8CctdRAmT9xDDdaAFPlu5CAYhgTl2xzKM4+WKO4AWL2z8xMWqOuTk5C5vECU0jsxMceSp7XgIBlECJhTIadn32Z6jmh6gxQf+zIUYdv7Q0vUCBiGBRnSPKubKwVbTHrIIk+tyFCfmhW0X4RNRgtwWIr/D5K077MlDlJBq4vo9E9fDAO/XIxlECWgY7nkzDQQGzLZLPEQJJUZWR3HATDwPwSCLMBkR+pIk/+N45B6iBOYPZH+UBGFzr296iBJalhtZ5OaB2bel5CFK4D2pGCIiLn1c7WCQRRh9Dv0duBpblXuIEiht69UYhXkIBlECvnHXr82jg/FqaYOQQNM0MZ+yNM8JoHEALY5eQkUtzd9zu8obRAncix4yRiIT6I9DMIgSckwM7GAXhrN0OoaDLMKUq4o4SPCp01WDQZRQqqzA8gKEWsvpGA6ihMosEpniFus+xeoiSuDMFKoqht7udzCIEjqahMOQmiefBUsPUcJMcrsdbzKA2XN6koOQwKSjnWYl73MuvE/iITiIEjJPTBVq28GMxOMQDKKEVjo9x66nCo4KcBAlzDyZGpE3lqa877txERI6HRYulGDqgNpte5r2ECVIqqUqkx5655msHIQELvrA9OovxrrP3k/XcBASJpcwQpZIX5423iFuHgICk2Fz3TO/5JKznPdRVh8ipaMbD3ZehrRgWgp7DLkQKVwr0PUMJofkRVPPOHWhH75KoHHUuSyGQQ8LJpTn6/oQKfGdSpZLatNE1yk7ItCHlMLdGEapMAVYzOl5MB9SyhhcP5R8T8yEEw/DICTQOWti4GdGeZ4NQB9SSmYcfZP1n57THh4+pBTuJTLZSBaveF+w6ENK6dzJSSJv9aQu8SGljMi9GZFjNtxp+X1IKVB6c+jCVGT6wHYoFiIlc/d8iRP3bPrDcBAlxFQlO6vc6Uxb+jAspBR0uCFuBM2xvq839CGl8FbIqVFWPe7QQhdZhDF4yYqIoV+f3S4fUgojYWTjuWlO1MMwiBIqxlzST8KlzHE9lYWUAlMvVpXPvLdaXEQJeKcyl5jBqxfDQkphlAFDmihnIMzVhhZaFFj4Ma+fqr3fFAMpBQ6nXBpNean7XL0PkVJoR+zaIwf2Q/EgpTAGL+d1PTHjRw7FQkqhuZtWh+B81g7FQkoBW/Jq8QJwPPHpLB6klALnLM7VUffV2C6yCLPrgVcOhlZO9/IgpTCsuKo8x/OBLaDFG/Mk6TfkCmkph2AhpQxoGJ7FzLy2dMzT7h5ECky6hmlZPfo0w74y0IeUAtdLcnEzPCMxCuZQLKQUnjumf07tXMW73xQLKQUDWXIFQR4mD20fioVIYS71wNwpko21xdNTHEQJlbmLp6w04BfHGIdhIVKYPg8Gj86YTBbc9pt4EClD1qK4NsHbW2EX70ochATZzGVGHjZB4P7OQ3AQECSZPq8WfvH6KwZRPNaND5HSJVBXDpHzucO+99GHSJkMl0K35uU6vGP1eSoH0OKzdB5D1NOr+0pUFwEh8VKwIKYeNWY9T+QhSmglczamx8GduXoIBiGBOcFgaJfX4EU2MEOfDuIhSmD22TZpfaJbP36jB2hx2J/MOTaYuinuk9guooQWZP10cIcitnxVYBAlDJpfkeIw884P5yJK4AUzqE4Wlvq3RzIICTSzeeSD4pa5ZPUQHEQJEVq0SBBSQdvsvuchSki8cV5+Byr/WRTxAC3OzIhZXIbIGIDrgQyiBLR902qZFXsrWg9RQoUOaSoOZZ/5cREltMBbHuVQH2aEcT2SQRaBR1W6rKDNEMr1zgZRQueZryxieJ+jHoJBlDC4BCrnDEc+V2G5iBJ47b0eTKTHVK+XNsgioHEwauV30LDtInwiJIhuhs3UV2KeXd4CWpw3dfb1JeC51VPeIEpIRa4DNa3gIEqAe8t7Y9ma6C17QvUQJRR+ui4djHffjUMwiBJq4VUAFKe9A+DItTDcoqzCPHmB6CluECVgSmGOVIhry3vL2kWUwDu/g3RfdPbQ0iEYRAlM/iQ/w9vunhQXHsDiNPOk/QY38kc4vdRBlJCK3MMpWk1c4k0wiBIwhTHbD1etE7PhHIJBlNCZ+rBRkUcepblqMAgJVc7G0P3nlV11+6AOoMVhzjE0dzD4Ku1MbC6ihMG958jJjmF87arAICTQ7alM7sHbskI8ppKHkDDe3OdDD4y84yiVeKrwIFAy4wK5cvGKcpaBdwAsiguRAqNLtlJ5QRXM+bRnZw8hgYHyeFpeBMXzRHlf/uNDpDAtEgwomGmRGY/aDlH0IVImJ+7G4AOqlR6PmnEhpfDSFUaKJwnoac8+gQ8ppc/J9LxJj1gcF9eFQCm8MiEx+I8GJON6N8WFlAItlJlSMk0uoYbte7uQUiqGMVOpMNC6pn3EyoeU0rn3PCWEhjvl+TAMogSMmiJuVtAbui+GhUihkZ3ldifec1BzeoLufUgpsZZHznvD9hzpQkqBY1aLWvWYdra68hAlZAb0qpi3huXrsSykFIaSTj1C1m9X2oWUUuEtTf2pUcPYi0EupJQmq2oin8yLdFEspJTOzJJLPsdo1/ey0KJM7vzpOlytO8e/DyllTLkzXOSYtfeqngspZZase9eMjIC9cT2YhRZlymaNyCNmi5tiIFJojHILdsmvDuYgizBHkYN9rHvs0NVfvP7wSq//9gqv371gCb/++Pqfr7/H3/7x9dO/+c2///Ovf/PLn//s9esfvxjfBr8QWity6YD5KfDDR8p0EhJ2vXrhT/86vn77I37oj/j35/cT9Hf6/kg8oVmYDVf61u9BTUKVpyH5qpu5YBsj8b//xCX3aj8/cNX1Z5/hF1//HwNru+sKZW5kc3RyZWFtCmVuZG9iagoxMSAwIG9iagoxMzQxNwplbmRvYmoKMTYgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAzMjcgPj4Kc3RyZWFtCnicPZJLjsQwCET3OQUXaMn8/DlPRrPquf+2H3Z6FlGhAEVRuGtKk+jyUpccKtmb/OgVU8XN5O+JhsQ0cTfQwSn3taMI/gS4DmbuDNFKiUG9dYnK8pGx89fX05cH78vbYbBRepaYV5+SsQYL8nR08QHm3Nruf5XvK5OOLL1KT0XvS71YlqgPMfti9SncxuYb23ownkzxazZRq5lT1toiugzURUo3sdULUO1RgeNcR28VbQMQgTtBB5UJtWpswUhItEXWL8xpQvfE/+0Bul/axHXsg9i0jWd8RRpD0N24R1nDzDHqfGpzw9rT2SbrlOJMLteyOFYJNErLWGpL8Kx6XRRjgxlopRDN0WpsZWh26OtleD/IDG87irGopUrPnjen4Fx97NcUevA8ix3Bm5zYLejnkKxCBi02YeosXa/wOed9/X4AKj19RgplbmRzdHJlYW0KZW5kb2JqCjE3IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzI1ID4+CnN0cmVhbQp4nDWSuXFDMRBD81/FNuAZ7sWjHnkc2f2nfqDsQAOI5IIAPjvchrnbh6f1GNZz2Kc/l6bbz1M7rWra91N9rHpbJSsTHMtqtb2e3G7FL7vAtox58fXE2pdFcWItC+fEdHZ8M11tvliJvJhHM5fNtBgo5baYB0W/amnpxwKldHbWuYiDsS3HwWXiLE7gTvvSBntypgqGnjIEcYW+Qztiwdlsk/QmqO7LohQLzqieQFEoh+cyL6H6W/Yh6yQdxuJWpTFk+UATX4Qm3SlDzekHawhx8aSDDTDgFxiacf+fZWuQvQ1H2GlGFXRO84w33oHLCJ05UKFmRcwlKdxEqhipRy2Yr+AExvJdkezHkZ8z6ROLADOCpRAQ1eh/H9zbeSty5ERqSgo+vF6O7hfK0Zupy27muKP7ll2akVKAw/6f2uv5+gWYMXm7CmVuZHN0cmVhbQplbmRvYmoKMTggMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxNzggPj4Kc3RyZWFtCnicPZBLEgMhCET3nqKPID/R8ySV1eT+2zTOmIX2EyhssKXoGM7L1ZBd8ZZWGJ74Nu8LnomrqfWHJBUy+6YOGYtn8hQnJBSvJmNA3LHV1qNxMsIMuywmZmCuiq9ELqhQAupR8mpmo+BqpoK+fcRWmfUWFwhFAiYsZyv+nwPT6xYdDBaY7TfLszz2CtN0LMx7hnkPRSN+BuVabmBlrYOfhh2a97ZoKP/kJ3sWeLXPD96rQqEKZW5kc3RyZWFtCmVuZG9iagoxNCAwIG9iago8PCAvQmFzZUZvbnQgL0RlamFWdVNhbnMtT2JsaXF1ZSAvQ2hhclByb2NzIDE1IDAgUgovRW5jb2RpbmcgPDwgL0RpZmZlcmVuY2VzIFsgMTAxIC9lIDExNSAvcyAvdCBdIC9UeXBlIC9FbmNvZGluZyA+PgovRmlyc3RDaGFyIDAgL0ZvbnRCQm94IFsgLTEwMTYgLTM1MSAxNjYwIDEwNjggXSAvRm9udERlc2NyaXB0b3IgMTMgMCBSCi9Gb250TWF0cml4IFsgMC4wMDEgMCAwIDAuMDAxIDAgMCBdIC9MYXN0Q2hhciAyNTUgL05hbWUgL0RlamFWdVNhbnMtT2JsaXF1ZQovU3VidHlwZSAvVHlwZTMgL1R5cGUgL0ZvbnQgL1dpZHRocyAxMiAwIFIgPj4KZW5kb2JqCjEzIDAgb2JqCjw8IC9Bc2NlbnQgOTI5IC9DYXBIZWlnaHQgMCAvRGVzY2VudCAtMjM2IC9GbGFncyA5NgovRm9udEJCb3ggWyAtMTAxNiAtMzUxIDE2NjAgMTA2OCBdIC9Gb250TmFtZSAvRGVqYVZ1U2Fucy1PYmxpcXVlCi9JdGFsaWNBbmdsZSAwIC9NYXhXaWR0aCAxMzUwIC9TdGVtViAwIC9UeXBlIC9Gb250RGVzY3JpcHRvciAvWEhlaWdodCAwID4+CmVuZG9iagoxMiAwIG9iagpbIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwCjYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgMzE4IDQwMSA0NjAgODM4IDYzNgo5NTAgNzgwIDI3NSAzOTAgMzkwIDUwMCA4MzggMzE4IDM2MSAzMTggMzM3IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYKNjM2IDYzNiAzMzcgMzM3IDgzOCA4MzggODM4IDUzMSAxMDAwIDY4NCA2ODYgNjk4IDc3MCA2MzIgNTc1IDc3NSA3NTIgMjk1CjI5NSA2NTYgNTU3IDg2MyA3NDggNzg3IDYwMyA3ODcgNjk1IDYzNSA2MTEgNzMyIDY4NCA5ODkgNjg1IDYxMSA2ODUgMzkwIDMzNwozOTAgODM4IDUwMCA1MDAgNjEzIDYzNSA1NTAgNjM1IDYxNSAzNTIgNjM1IDYzNCAyNzggMjc4IDU3OSAyNzggOTc0IDYzNCA2MTIKNjM1IDYzNSA0MTEgNTIxIDM5MiA2MzQgNTkyIDgxOCA1OTIgNTkyIDUyNSA2MzYgMzM3IDYzNiA4MzggNjAwIDYzNiA2MDAgMzE4CjM1MiA1MTggMTAwMCA1MDAgNTAwIDUwMCAxMzUwIDYzNSA0MDAgMTA3MCA2MDAgNjg1IDYwMCA2MDAgMzE4IDMxOCA1MTggNTE4CjU5MCA1MDAgMTAwMCA1MDAgMTAwMCA1MjEgNDAwIDEwMjggNjAwIDUyNSA2MTEgMzE4IDQwMSA2MzYgNjM2IDYzNiA2MzYgMzM3CjUwMCA1MDAgMTAwMCA0NzEgNjE3IDgzOCAzNjEgMTAwMCA1MDAgNTAwIDgzOCA0MDEgNDAxIDUwMCA2MzYgNjM2IDMxOCA1MDAKNDAxIDQ3MSA2MTcgOTY5IDk2OSA5NjkgNTMxIDY4NCA2ODQgNjg0IDY4NCA2ODQgNjg0IDk3NCA2OTggNjMyIDYzMiA2MzIgNjMyCjI5NSAyOTUgMjk1IDI5NSA3NzUgNzQ4IDc4NyA3ODcgNzg3IDc4NyA3ODcgODM4IDc4NyA3MzIgNzMyIDczMiA3MzIgNjExIDYwOAo2MzAgNjEzIDYxMyA2MTMgNjEzIDYxMyA2MTMgOTk1IDU1MCA2MTUgNjE1IDYxNSA2MTUgMjc4IDI3OCAyNzggMjc4IDYxMiA2MzQKNjEyIDYxMiA2MTIgNjEyIDYxMiA4MzggNjEyIDYzNCA2MzQgNjM0IDYzNCA1OTIgNjM1IDU5MiBdCmVuZG9iagoxNSAwIG9iago8PCAvZSAxNiAwIFIgL3MgMTcgMCBSIC90IDE4IDAgUiA+PgplbmRvYmoKMjMgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyMjcgPj4Kc3RyZWFtCnicRZBLjgMhEEP3nMJHoP5wno6y6tx/Oy460WywJSjXM7kDE3vxyEyUbrxkuDvMFZ/hGTBz+EqYbMRM6E5cI5SuFOFUnwiOy9686aCJe0TOo54FWdXODWoBZ5Lmgu2CLueMFTfOya1G5c30o9cQvml3H9d5h/ZI497DVL7OOdXOu4oIaQtZ1bS+kGwUyQ5rPypN287LELHgbBpW6BzeCLtXdydLGH6/8jAlEz8P3fpRfF03o+sUZydXJmvPeehJ6V/1yqMPxXEp6Bex7cwk91+nU5KpU3tn72n9p7jH+w+QblY5CmVuZHN0cmVhbQplbmRvYmoKMjQgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyNzkgPj4Kc3RyZWFtCnicPVI7ksUgDOtzCl+AGfAXzpOdrbL3b1eCvFdkrERGkh3SUroMDWkjVWIMqezyMy7CtlT+DiL7XD6/0NY60JyHo4uNCaAp96WrA3YVrZIWvusoMgTmENhommhOUXAKCbV92nCQh/oA6zIidr0vBiV6NrKtQ/Yg9hOpGVJBOfCOhFCEh82+Y/g4cTztzRNgTp4MOCN79TNOKV05YTkUMDK3wy2w3tdnX89GuYi85gdhDiLHInJiUxMMkkCxHF6OdPDUU6M2A+RRXDBRx2yxwM+9Zz7oMaRT6JHt2FwupF5k8IX52aHwYD+3AAYqqrWdBjtYvb+OY72GDb+P4VrkjtuKLRygTRcOxDvBEVnv7y15rt9/B3JuBgplbmRzdHJlYW0KZW5kb2JqCjI1IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjc2ID4+CnN0cmVhbQp4nD1SS47oMAjb5xRcIFL4Jj1PR7Pqu//22bSaFRZgY0jKS5aopUwtk1SVXUt+dHTuMvk3NK9Gz7D15sxD5kmxODK3yj0M5JlbrNBrb9SrCwCxSO7UEtslxpZLxZJkh6ztEHdk0OHFKtnM+NrCDjdrjkdSFypepwcF+Yww8E7Mfb6JZSE0Vxltt8BBDxaoAz0wCxO5IuPdaxM9jXgLoKwPmb+ILjfUXI9sTIBTuNoL3qmU1jEPFYmiHOxGGr2fo+Y4ocPbPWI59nIJRwbe4mjHe/BFiJ5GVkSsfsizEfmGDalIj5jKG+puH3r8ja3YaH92JknB11/W20w10rng1Fyt1l+DR2jw3qjh8/dLnvH7HyEmbhIKZW5kc3RyZWFtCmVuZG9iagoyNiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDI0NSA+PgpzdHJlYW0KeJxFULuNQzEM6z0FFwhg/Sx7nndIldu/PUpGcIUhWj+SWhKYiMBLDLGUb+JHRkE9C78XheIzxM8XhUHOhKRAnPUZEJl4htpGbuh2cM68wzOMOQIXxVpwptOZ9lzY5JwHJxDObZTxjEK6SVQVcVSfcUzxqrLPjdeBpbVss9OR7CGNhEtJJSaXflMq/7QpWyro2kUTsEjkgZNNNOEsP0OSYsyglFH3MLWO9HGykUd10MnZnDktmdnup+1MfA9YJplR5Smd5zI+J6nzXE597rMd0eSipVX7nP3ekZbyIrXbodXpVyVRmY3Vp5C4PP+Mn/H+A46gWT4KZW5kc3RyZWFtCmVuZG9iagoyNyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDI0NyA+PgpzdHJlYW0KeJxNUbttRDEM698UXOAA62t5ngtSXfZvQ8kIkMIgoS8ppyUW9sZLDOEHWw++5JFVQ38ePzHsMyw9yeTUP+a5yVQUvhWqm5hQF2Lh/WgEvBZ0LyIrygffj2UMc8734KMQl2AmNGCsb0kmF9W8M2TCiaGOw0GbVBh3TRQsrhXNM8jtVjeyOrMgbHglE+LGAEQE2ReQzWCjjLGVkMVyHqgKkgVaYNfpG1GLgiuU1gl0otbEuszgq+f2djdDL/LgqLp4fQzrS7DC6KV7LHyuQh/M9Ew7d0kjvfCmExFmDwVSmZ2RlTo9Yn23QP+fZSv4+8nP8/0LFShcKgplbmRzdHJlYW0KZW5kb2JqCjI4IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggOTAgPj4Kc3RyZWFtCnicTY1BEsAgCAPvvCJPUETQ/3R60v9fq9QOvcBOAokWRYL0NWpLMO64MhVrUCmYlJfAVTBcC9ruosr+MklMnYbTe7cDg7LxcYPSSfv2cXoAq/16Bt0P0hwiWAplbmRzdHJlYW0KZW5kb2JqCjI5IDAgb2JqCjw8IC9CQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM3Ci9TdWJ0eXBlIC9Gb3JtIC9UeXBlIC9YT2JqZWN0ID4+CnN0cmVhbQp4nOMyNDBTMDY1VcjlMjc2ArNywCwjcyMgCySLYEFk0wABXwoKCmVuZHN0cmVhbQplbmRvYmoKMzAgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA4MCA+PgpzdHJlYW0KeJxFjLsNwDAIRHumYAR+JmafKJWzfxsgStxwT7p7uDoSMlPeYYaHBJ4MLIZT8QaZo2A1uEZSjZ3so7BuX3WB5npTq/X3BypPdnZxPc3LGfQKZW5kc3RyZWFtCmVuZG9iagozMSAwIG9iago8PCAvRmlsdGVyIC9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      "text/plain": [
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     "metadata": {},
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   ],
   "source": [
    "y = x.subs({omega: 10, sigma: -.1})\n",
    "\n",
    "sym.plot(sym.re(y), (t, 0, 2*sym.pi), ylabel=r'Re{$e^{st}$}')\n",
    "sym.plot(sym.im(y), (t, 0, 2*sym.pi), ylabel=r'Im{$e^{st}$}');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Try other values for `omega` and `sigma` to create signals with increasing/constant/decreasing amplitudes and different angular frequencies."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dirac Impulse\n",
    "\n",
    "The Dirac impulse is one of the most important signals in the theory of signals and systems. It is used for the characterization of LTI systems and the modeling of impulse-like signals. The Dirac impulse is defined by way of the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) which is not a function in the conventional sense. It is a generalized function or *distribution*. The Dirac impulse is denoted as $\\delta(t)$. The Dirac delta function is defined by its effect on other functions. A rigorous treatment is beyond the scope of this course material. Please refer to the literature for a detailed discussion of the mathematical foundations of the Dirac delta distribution. Fortunately it is suitable to consider only certain properties for its application in signal processing. The most relevant ones are\n",
    "\n",
    "1. **Sifting property**\n",
    "    \\begin{equation}\n",
    "    \\int_{-\\infty}^{\\infty} \\delta(t) \\cdot x(t) = x(0)\n",
    "    \\end{equation}\n",
    "    where $x(t)$ needs to be differentiable at $t=0$. The sifting property implies $\\int_{-\\infty}^{\\infty} \\delta(t) = 1$.\n",
    "    \n",
    "2. **Multiplication**\n",
    "    \\begin{equation}\n",
    "    x(t) \\cdot \\delta(t) = x(0) \\cdot \\delta(t)\n",
    "    \\end{equation}\n",
    "    where $x(t)$ needs to be differentiable at $t=0$.\n",
    "    \n",
    "3. **Linearity**\n",
    "    \\begin{equation}\n",
    "    a \\cdot \\delta(t) + b \\cdot \\delta(t) = (a+b) \\cdot \\delta(t)\n",
    "    \\end{equation}\n",
    "    \n",
    "4. **Scaling**\n",
    "    \\begin{equation}\n",
    "    \\delta(a t) = \\frac{1}{|a|} \\delta(t)\n",
    "    \\end{equation}\n",
    "    where $a \\in \\mathbb{R} \\setminus 0$. This implies that the Dirac impulse is a function with even symmetry.\n",
    "    \n",
    "5. **Derivation**\n",
    "    \\begin{equation}\n",
    "    \\int_{-\\infty}^{\\infty} \\frac{d \\delta(t)}{dt} \\cdot x(t) \\; dt = - \\frac{d x(t)}{dt} \\bigg\\vert_{t = 0}\n",
    "    \\end{equation}\n",
    "\n",
    "6. **Convolution**\n",
    "    \n",
    "    Generalization of the sifting property yields\n",
    "    \\begin{equation}\n",
    "    \\int_{-\\infty}^{\\infty} \\delta(\\tau) \\cdot x(t - \\tau)  \\, d\\tau = x(t)\n",
    "    \\end{equation}\n",
    "    \n",
    "    This operation is known as [convolution](https://en.wikipedia.org/wiki/Convolution) and will be introduced later in more detail. It may be concluded already here that the Dirac delta function constitutes the neutral element of the convolution.\n",
    "\n",
    "It is important to note that the product $\\delta(t) \\cdot \\delta(t)$ of two Dirac impulses is not defined."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "This example illustrates some of the basic properties of the Dirac impulse. Let's first define a Dirac impulse by way of the Dirac delta function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\delta\\left(t\\right)$"
      ],
      "text/plain": [
       "δ(t)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "delta = sym.DiracDelta(t)\n",
    "delta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's check the sifting property by defining an arbitrary signal (function) $f(t)$ and integrating over its product with the Delta impulse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAVCAYAAADM+lfpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAChUlEQVRIDaWW7VEbQQyGHSYFeKCDowMHKsDugLiDOB0kk1/2P8Z0AB1kSAemA4g7wB3Ekw6c91lWh25PdzlAM7K0+nhPK+3denQ4HEbv5eVyWb0WI8o5Gr2TVqvVN0FM3gBT5dw69QM76iIF38j3N/uvtDY9mbS+lHIu+T3H1EK2dV78kTwVr2Xb1QFStGYjO8lf2Ds7owDAjsU/xQvxVFyT/GMtfkhGhfyW7wGf+Fo6MRvplWRN2QcGWN3FyEcBFEJBj+J7sSeKpXMNEjB5Y8m0W5zS6SjrVny2gRUXo2S6QLX30uFZBiTHaC7brS2c/Cx969amPkiZKid1wYwZA6xx15hmCmaW7KhFslNsY/4uCN/erU01rMa4sxOs+UeLROohtJhCEqDWd9L3kl8lPbH7cmzkN3btE5zO2EsCa1YWQ9tvBcorxtvD4YvoTMZo/vYg64LPtW5FBT8pcNoakwqwE9/auUMG0MCdeZB6EkSBFZ6ZCcEqahskmYkO9O3e4ry0rvHdKYkzU7U6I+O5uOtwliCNtTZgBUajMFuEnTYXFUNn+rpCAQAaOGtPjNdG7e3WmWj8YO2jYjicfBP6iBlHDySHNxCMkj7JsHXd834K3TWKUSAPoMqoep9M5xhni4TBG8nngHsrkXQw5+Ivz5bWL4VuGq+2DIxopOT/jWmjsOjVJh0CnDvHukfhFz24fNfWjf8y+o9xI74b8t9EcU/iyZDYvhhhVGARc6RqL8XMGWLWXI5DiMut/DIPyStjuNHrixKFWza9HZL1bVtm+bXiOBv8QbJRePcgPeeCkS5czkwqJkvunNcQ8XSV++wtxLl7eWbfPIf48swXQ2J9jPIW5HrbPwk65+Kh5LaYAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle f{\\left(0 \\right)}$"
      ],
      "text/plain": [
       "f(0)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sym.Function('f')(t)\n",
    "sym.integrate(delta*f, (t, -sym.oo, sym.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Derive the sifting property for a shifted Dirac impulse $\\delta(t-\\tau)$ and check your results by modifying above example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Heaviside Signal\n",
    "\n",
    "The Heaviside signal is defined by the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function)\n",
    "\n",
    "\\begin{equation}\n",
    "\\epsilon(t) = \\begin{cases} 0 & t<0 \\\\  \\frac{1}{2} & t=0 \\\\ 1 & t > 0 \\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "Note that alternative definitions exist, which differ with respect to the value of $\\epsilon(t)$ at $t=0$. The Heaviside signal may be used to represent a signal that switches on at a specified time and stays switched on indefinitely. The Heaviside signal can be related to the Dirac impulse by\n",
    "\n",
    "\\begin{equation}\n",
    "\\epsilon(t) = \\int_{-\\infty}^{t} \\delta(\\tau) \\; d\\tau\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "In the following, a Heaviside signal $\\epsilon(t)$ is defined and plotted. Note that `Sympy` denotes the Heaviside function by $\\theta(t)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "θ(t)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step = sym.Heaviside(t)\n",
    "step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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       "    <path d=\"M 19.341406 15.8 \n",
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       "   <rect height=\"315.130051\" width=\"419.184375\" x=\"19.341406\" y=\"15.8\"/>\n",
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       " </defs>\n",
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      ],
      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(step, (t, -2, 2), ylim=[-0.2, 1.2], ylabel=r'$\\epsilon(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's construct a harmonic signal $\\cos(\\omega t)$ with $\\omega=2$ which is switched on at $t=0$. Considering the definition of the Heaviside function, the desired signal is given as \n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = \\cos(\\omega t) \\cdot \\epsilon(t)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   ],
   "source": [
    "x = sym.cos(omega*t) * sym.Heaviside(t)\n",
    "sym.plot(x.subs(omega,2), (t, -2, 10), ylim=[-1.2, 1.2], ylabel=r'$x(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "### Rectangular Signal\n",
    "\n",
    "The rectangular signal is defined by the [rectangular function](https://en.wikipedia.org/wiki/Rectangular_function)\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{rect}(t) = \\begin{cases} 1 & |t| < \\frac{1}{2} \\\\ \\frac{1}{2} & |t| = \\frac{1}{2} \\\\ 0 & |t| > \\frac{1}{2} \\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "Its time limits and amplitude are chosen such that the area under the function is $1$.\n",
    "\n",
    "Note that alternative definitions exist, which differ with respect to the value of $\\text{rect}(t)$ at $t = \\pm \\frac{1}{2}$. The rectangular signal is used to represent a signal which has finite duration, respectively is switched on for a limited period of time. The rectangular signal can be related to the Heaviside signal by\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{rect}(t) = \\epsilon \\left(t + \\frac{1}{2} \\right) - \\epsilon \\left(t - \\frac{1}{2} \\right)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "**Example**\n",
    "\n",
    "The Heaviside function is used to define a rectangular function in `Sympy`. This function is then used as rectangular signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "class rect(sym.Function):\n",
    "\n",
    "    @classmethod\n",
    "    def eval(cls, arg):\n",
    "        return sym.Heaviside(arg + sym.S.Half) - sym.Heaviside(arg - sym.S.Half)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(rect(t), (t, -1, 1), ylim=[-0.2, 1.2], ylabel=r'rect$(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Use $\\text{rect}(t)$ to construct a harmonic signal $\\cos(\\omega t)$ with $\\omega=2$ which is switched on at $t=-\\frac{1}{2}$ and switched off at $t=+\\frac{1}{2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sign Signal\n",
    "\n",
    "The sign signal is defined by the [sign/signum function](https://en.wikipedia.org/wiki/Sign_function) which evaluates the sign of its argument\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{sgn}(t) = \\begin{cases} 1 & t>0 \\\\ 0 & t=0 \\\\ -1 & t < 0  \\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "The sign signal is useful to represent the absolute value of a real-valued signal $x(t) \\in \\mathbb{R}$ by a multiplication\n",
    "\n",
    "\\begin{equation}\n",
    "|x(t)| = x(t) \\cdot \\text{sgn}(x(t))\n",
    "\\end{equation}\n",
    "\n",
    "It is related to the Heaviside signal by\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{sgn}(t) = 2 \\cdot \\epsilon(t) - 1\n",
    "\\end{equation}\n",
    "\n",
    "when following above definition with $\\epsilon(0)=\\frac{1}{2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following example illustrates the sign signal $\\text{sgn}(t)$. Note that the sign function is represented as $\\text{sign}(t)$ in `Sympy`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\operatorname{sign}{\\left(t \\right)}$"
      ],
      "text/plain": [
       "sign(t)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgn = sym.sign(t)\n",
    "sgn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "sym.plot(sgn, (t, -2, 2), ylim=[-1.2, 1.2], ylabel=r'sgn$(t)$');"
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   "source": [
    "**Exercise**\n",
    "\n",
    "* Check the values of $\\text{sgn}(t)$ for $t \\to 0^-$, $t = 0$ and $t \\to 0^+$ as implemented in `SymPy`. Do they conform to above definition?"
   ]
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*."
   ]
  }
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